Please show step by step Thank you 6.21 Infant deaths in King County, Washington
ID: 3329663 • Letter: P
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6.21 Infant deaths in King County, Washington were grouped by season of the year. The number of deaths by season, for selected causes of death, are listed in Table 6.13. Table 6.13 Death Data for Problem 6.21 Season Winter Spring Summer Autumn 50 30 95 40 48 40 93 19 71 34 35 83 43 86 46 36 Asphyxia Immaturity Congenital malformations Infection Sudden infant death syndrome78 40 87 (a) At the 5% significance level, test the hypothesis that SIDS deaths are uniformly (b) At the 10% significance level, test the hypothesis that the deaths due to infection (c) What can you say about the p-value for testing that asphyxia deaths are spread (p 1 /4) spread among the seasons are uniformly spread among the seasons uniformly among seasons? Immaturity deaths?Explanation / Answer
Part (a)
To test SIDS deaths are uniformly distributed among seasons, test statistic is:
2 = [i = 1,4]{(Oi - Ei)2/Ei}which can also be put in a simplified form as:
2 = [i = 1,4](Oi2/Ei) – n, where Oi and Ei are respectively the observed (given) and expected number of SIDS deaths during season i, i = 1 for winter, 2 for spring, 3 for summer and 4 for autumn and n = total number of SIDS deaths.
Further, under the hypothesis that SIDS deaths are uniformly distributed among seasons,
E1 = E2 = E3 = E4 = (¼)n = E, say. Thus, the test statistic simplifies itself into:
2 = {(4/n)[i = 1,4]Oi2} – n
Calculations:
i
1
2
3
4
Total
Oi
78
71
87
86
322
Oi^2
6084
5041
7569
7396
26090
4/n
0.0124224
Chi-square
2.0993789
Under the hypothesis, 2 is distributed as Chi-square with (k - 1) degrees of freedom, where k is the number of comparisons made [k = 4 in the given question.]
Given level of significance, = 5%, the critical value = upper % point of 23 = 7.81.
Since calculated value (2.10) < critical value (7.81), the hypothesis is accepted.
=> there is sufficient evidence to suggest that SIDS deaths are uniformly distributed among seasons. ANSWER
Part (b)
To test if deaths due to infection are uniformly distributed among seasons, the procedure is identical to what was done above under Part (a)
Calculations:
i
1
2
3
4
Total
Oi
40
19
40
43
142
Oi^2
1600
361
1600
1849
5410
4/n
0.028169014
Chi-square
10.3943662
Given level of significance, = 10%, the critical value = upper % point of 23 = 6.25.
Since calculated value (10.39) > critical value (6.25), the hypothesis is rejected.
=> there is not sufficient evidence to suggest that deaths due to infection are uniformly distributed among seasons. ANSWER
Part (c)(i)
To find p-value for testing the hypothesis that asphyxia deaths are uniformly distributed among seasons, the procedure is identical to what was done above under Parts (a) and (b).
Calculations
i
1
2
3
4
Total
Oi
50
48
46
34
178
Oi^2
2500
2304
2116
1156
8076
4/n
0.02247191
Chisquare
3.483146067
p-value = P(23 > 3.48) = 0.3234 ANSWER [using Excel Function of Chi-square distribution]
Part (c)(ii)
Identically, for deaths due to immaturity,
Calculations
i
1
2
3
4
Total
Oi
30
40
36
35
141
Oi^2
900
1600
1296
1225
5021
4/n
0.028368794
Chisquare
1.439716312
p-value = P(23 > 1.43) = 0.6985 ANSWER [using Excel Function of Chi-square distribution]
i
1
2
3
4
Total
Oi
78
71
87
86
322
Oi^2
6084
5041
7569
7396
26090
4/n
0.0124224
Chi-square
2.0993789
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